Design and design check for multi layer well systems MWell
MWELL Design and design check for multi layer well systems Verification Report Version: 15.1 Revision: 44405 16 March 2017
MWELL, Verification Report Published and printed by: Deltares Boussinesqweg 1 2629 HV Delft P.O. 177 2600 MH Delft The Netherlands telephone: +31 88 335 82 73 fax: +31 88 335 85 82 e-mail: info@deltares.nl www: https://www.deltares.nl For sales contact: telephone: +31 88 335 81 88 fax: +31 88 335 81 11 e-mail: sales@deltaressystems.nl www: http://www.deltaressystems.nl For support contact: telephone: +31 88 335 81 00 fax: +31 88 335 81 11 e-mail: support@deltaressystems.nl www: http://www.deltaressystems.nl Copyright 2017 Deltares All rights reserved. No part of this document may be reproduced in any form by print, photo print, photo copy, microfilm or any other means, without written permission from the publisher: Deltares.
Contents Contents List of Figures List of Tables iii v Introduction 1 1 Group 1: Benchmarks from literature (exact solution) 3 1.1 Huisman & Kemperman solution for a two layers system........... 3 1.1.1 Description.............................. 3 1.1.2 Benchmark results.......................... 3 1.1.3 MWELL results............................ 4 1.2 Hantush & Jacob solution for leaky aquifers.................. 6 1.2.1 Description.............................. 6 1.2.2 Benchmark results.......................... 6 1.2.3 MWell results............................. 7 2 Group 2: Benchmarks from literature (approximate solution) 9 3 Group 3: Benchmarks from spreadsheets 11 4 Group 4: Benchmarks generated by MWell 13 5 Group 5: Benchmarks compared with other programs 15 Bibliography 17 Deltares iii
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List of Figures List of Figures 1.1 Geometry of benchmark 1-1......................... 3 1.2 MWell results for benchmark 1-1a....................... 4 1.3 MWell results for benchmark 1-1b....................... 5 1.4 Geometry of benchmark 1-2......................... 6 Deltares v
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List of Tables List of Tables 1.1 Results of benchmark 1-1a: Case I...................... 5 1.2 Results of benchmark 1-1b: Case II...................... 5 1.3 Results of benchmark 1-2 at time 1, 2 and 5 days............... 7 1.4 Results of benchmark 1-2 at time 10, 20 and 50 days............. 7 Deltares vii
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Introduction Deltares Systems commitment to quality control and quality assurance has led them to develop a formal and extensive procedure to verify the correct working of all of their geotechnical engineering tools. An extensive range of benchmark checks have been developed to check the correct functioning of each tool. During product development these checks are run on a regular basis to verify the improved product. These benchmark checks are provided in the following sections, to allow the user to overview the checking procedure and verify for themselves the correct functioning of MWELL. The benchmarks are subdivided into five separate groups as described below. Group 1 (chapter 1) - Benchmarks from literature (exact solution) Simple benchmarks for which an exact analytical result is available from literature. Group 2 (chapter 2) - Benchmarks from literature (approximate solution) More complex benchmarks described in literature for which an approximate solution is known. Group 3 (chapter 3) - Benchmarks from spread sheets Benchmarks which test program features specific to MWELL. Group 4 (chapter 4) - Benchmarks generated by MWELL Benchmarks for which the reference results are generated using MWELL. Group 5 (chapter 5) - Benchmarks compared with other programs Benchmarks for which the results of MWELL are compared with the results of other programs. The number of benchmarks in group 1 will probably remain the same in the future. The reason for this is that they are very simple, using only the most basic features of the program. The benchmarks in group 2 are well documented in literature. There are no exact solutions available for these problems; however in the literature estimated results are available. When verifying the program, the results should be close to the results found in the literature. The number of benchmarks in groups 3, 4 and 5 will grow as new versions of the program are released. These benchmarks are designed so that (new) features specific to the program can be verified. The benchmarks are kept as simple as possible so that only one specific feature is verified from one benchmark to the next. As much as software developers would wish they could, it is impossible to prove the correctness of any non-trivial program. Re-calculating all the benchmarks in this report, and making sure the results are as they should be, proves to some degree that the program works as it should. Nevertheless, there will always be combinations of input values that will cause the program to crash or to produce wrong results. Hopefully by using the verification procedure the number of ways this can occur will be limited. The benchmarks are all described in sufficient detail for reproduction to be possible at any time. The information given is enough to be able to make the calculation. The input files can be found on CD-ROM or can be downloaded from our website www.deltaressystems.com. Deltares 1 of 20
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1 Group 1: Benchmarks from literature (exact solution) 1.1 Huisman & Kemperman solution for a two layers system 1.1.1 Description Formula Huisman & Kemperman (Huisman and Kemperman (1951)) is an analytical solution for a two layered system. Two benchmarks with the same geometry are made: Case I (bm1-1a): discharge Q 1 of 50 m 3 /day from first aquifer; Case II (bm1-1b): discharge Q 2 of 30 m 3 /day from second aquifer. The same geometry is used for both benchmarks (see Figure 1.1) with: Resistance c 1 = 300 days Transmissibility k 1 H 1 = 10 m 2 /day Resistance c 2 = 100 days Transmissibility k 2 H 2 = 20 m 2 /day Figure 1.1: Geometry of benchmark 1-1 1.1.2 Benchmark results According to Huisman & Kemperman (Huisman and Kemperman, 1951): For case I: Drop of load in first aquifer [1]: S 1 = Q 1 [ ( 1 (λ1 λ 2π k 1 H 1 λ 2 α 2 ) k 0 λ1 r ) ( + (α 2 λ 2 ) k 0 λ2 r )] 1 Drop of load in second aquifer [2]: S 2 = Q 1 [ α 2 ( λ 2π k 1 H 1 λ 2 k0 λ1 r ) ( + k 0 λ2 r )] 1 For case II: Drop of head in first aquifer [1]: S 1 = Q 2 β 1 [ ( λ 2π k 2 H 1 λ 2 k0 λ1 r ) ( + k 0 λ2 r )] 2 Drop of head in second aquifer [2]: S 2 = Q 2 [ ( 1 (α2 λ 2π k 2 H 1 λ 2 λ 2 ) k 0 λ1 r ) ( + (λ 1 α 2 ) k 0 λ2 r )] 2 where: 1 α 1 = = 3.333 10-4 k 1 H 1 c 1 1 α 2 = = 5 10-4 k 2 H 2 c 2 Deltares 3 of 20
MWELL, Verification Report 1 β 1 = = 10-3 k 1 H 1 c( 2 ) λ 1 = 0.5 α 1 + α 2 + β 1 + (α 1 + α 2 + β 1 ) 2 4 α 1 α 2 = 1.737 10 3 ( ) λ 2 = 0.5 α 1 + α 2 + β 1 (α 1 + α 2 + β 1 ) 2 4 α 1 α 2 = 9.593 10 5 k 0 is the modified Bessel function of the second kind and order zero; r is the distance to the well, in m. The analytical results from this formula are performed in a spreadsheet for different distances to the well. The calculated results are given in Table 1.1 and Table 1.2. 1.1.3 MWELL results The numerical values of the drawdown in both aquifers can be found in the *.weh file as shown in Figure 1.2 and Figure 1.3. Comparison between MWell and Huisman & Kemperman solution is excellent. Figure 1.2: MWell results for benchmark 1-1a 4 of 20 Deltares
Group 1: Benchmarks from literature (exact solution) Figure 1.3: MWell results for benchmark 1-1b Table 1.1: Results of benchmark 1-1a: Case I Distance Drawdown in first aquifer Drawdown in second aquifer Benchmark MWELL Error Benchmark MWELL Error [m] [m] [m] [%] [m] [m] [%] 5 1.64195 1.64195 0.00 0.34457 0.34457 0.00 10 1.12654 1.12654 0.00 0.33181 0.33181 0.00 15 0.84887 0.84887 0.00 0.31625 0.31625 0.00 20 0.66952 0.66952 0.00 0.29944 0.29944 0.00 40 0.32537 0.32537 0.00 0.23275 0.23275 0.00 50 0.24611 0.24611 0.00 0.20335 0.20335 0.00 100 0.09049 0.09049 0.00 0.10287 0.10287 0.00 500 0.00081 0.00081 0.00 0.00100 0.00100 0.00 Table 1.2: Results of benchmark 1-1b: Case II Distance Drawdown in first aquifer Drawdown in second aquifer Benchmark MWELL Error Benchmark MWELL Error [m] [m] [m] [%] [m] [m] [%] 5 0.20674 0.20674 0.00 0.66487 0.66487 0.00 10 0.19908 0.19908 0.00 0.50387 0.50387 0.00 15 0.18975 0.18975 0.00 0.41278 0.41278 0.00 20 0.17966 0.17966 0.00 0.35057 0.35057 0.00 40 0.13965 0.13965 0.00 0.21399 0.21399 0.00 50 0.12201 0.12201 0.00 0.17551 0.17551 0.00 100 0.06172 0.06172 0.00 0.07858 0.07858 0.00 500 0.00060 0.00060 0.00 0.00074 0.00074 0.00 Use MWell input files bm1-1a.wei and bm1-1b.wei to run this benchmark. Deltares 5 of 20
MWELL, Verification Report 1.2 Hantush & Jacob solution for leaky aquifers 1.2.1 Description For wells in a leaky aquifer an approximate solution valid for small values of time is known under the name of Hantush & Jacob (Hantush and Jacob (1955)). The solution assumes no storage in the aquitard and also accounts for partially penetrating wells. This benchmark considers two wells in a one layer system. A semi pervious layer with resistance c = 5000 days of an aquifer with a coefficient of transmissibility k H = 200 m 2 /day and a specific yield S = 25.03%. Both wells have the following characteristics: Well number 1: X co-ordinate: 0 m Z co-ordinate: 0 m Discharge: Q1 =100 m3/day Well number 2: X co-ordinate: 0 m Z co-ordinate: 10 m Discharge: Q2 =50 m3/day Figure 1.4: Geometry of benchmark 1-2 1.2.2 Benchmark results According to Hantush & Jacob solution (Hantush and Jacob (1955)), the drop of head is given by the next formula: h (r, t) = where: u = S 4kH t r 2 Q (u, 4π kh W r ) B r is the distance to the well, in m; t is the time, in days; B is the leakage length, in m: B = kh c = 200 5000 = 1000 m W is the logarithmic integral: W ( ) ( ) u, r B = 1 exp y r2 dy y 4B 2 y u (1.1) 6 of 20 Deltares
Group 1: Benchmarks from literature (exact solution) The analytical results from this formula are performed in a spreadsheet for different times and different distances to the wells. The calculated results are given in the tables below. 1.2.3 MWell results The numerical values of the drawdown at different monitoring points and different times can be found in the *.weh file. Comparison between MWell and Hantush & Jacob solution is very good. Table 1.3: Results of benchmark 1-2 at time 1, 2 and 5 days Benchmark MWELL Error Time [days]: 1 2 5 1 2 5 1 2 5 X [m] Z [m] [m] [m] [m] [m] [m] [m] [%] [%] [%] 0 0 0.594 0.635 0.690 0.594 0.635 0.689 0.00 0.00 0.15 0 1 0.360 0.401 0.456 0.360 0.401 0.455 0.00 0.00 0.22 0 2 0.310 0.351 0.405 0.309 0.350 0.405 0.32 0.29 0.00 0 3 0.283 0.324 0.378 0.282 0.323 0.378 0.35 0.31 0.00 0 4 0.266 0.307 0.361 0.266 0.307 0.361 0.00 0.00 0.00 0 5 0.255 0.296 0.351 0.255 0.296 0.350 0.00 0.00 0.29 0 6 0.250 0.291 0.345 0.250 0.290 0.345 0.00 0.34 0.00 0 7 0.249 0.290 0.344 0.249 0.290 0.344 0.00 0.00 0.00 0 8 0.255 0.296 0.350 0.254 0.295 0.349 0.39 0.34 0.29 Table 1.4: Results of benchmark 1-2 at time 10, 20 and 50 days Benchmark MWELL Error Time [days]: 10 20 50 10 20 50 10 20 50 X [m] Z [m] [m] [m] [m] [m] [m] [m] [%] [%] [%] 0 0 0.731 0.772 0.825 0.730 0.771 0.824 0.14 0.13 0.12 0 1 0.497 0.538 0.591 0.496 0.537 0.590 0.20 0.19 0.17 0 2 0.446 0.487 0.540 0.445 0.486 0.540 0.22 0.21 0.00 0 3 0.419 0.460 0.513 0.419 0.459 0.513 0.00 0.22 0.00 0 4 0.402 0.443 0.497 0.402 0.443 0.496 0.00 0.00 0.20 0 5 0.392 0.433 0.486 0.391 0.432 0.485 0.26 0.23 0.21 0 6 0.386 0.427 0.480 0.386 0.426 0.480 0.00 0.23 0.00 0 7 0.386 0.426 0.480 0.385 0.426 0.479 0.26 0.00 0.21 0 8 0.391 0.432 0.485 0.390 0.431 0.484 0.26 0.23 0.21 Use MWell input file bm1-2.wei to run this benchmark. Deltares 7 of 20
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2 Group 2: Benchmarks from literature (approximate solution) This chapter contains benchmarks described in literature, for which an approximate solution is known (group 2). Deltares 9 of 20
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3 Group 3: Benchmarks from spreadsheets This chapter contains benchmarks which test program features specific to MWELL using spreadsheets (group 3). Deltares 11 of 20
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4 Group 4: Benchmarks generated by MWell This chapter contains benchmarks for which the reference results are generated using MWELL. Deltares 13 of 20
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5 Group 5: Benchmarks compared with other programs This chapter contains benchmarks for which the results of MWELL are compared with the results of other programs. Deltares 15 of 20
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Bibliography Hantush, M. S. and C. E. Jacob, 1955. Non-steady radial flow in an infinite leaky aquifer. Am. Geophys. Union Trans. vol. 36: pp. 95-100. Huisman, L. and J. Kemperman, 1951. Bernaling van spanningsgrondwater. De Ingenieur vol. 62 B13: pp. 29-35. Deltares 17 of 20
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